Because of friction and air resistance, each swing of a pendulum is a little shorter than the previous one. The lengths of the swings form a geometric sequence. Suppose the first swing of the pendulum has an arc length of 100cm and a return swing of 99cm.

a.)On which swing will the length first have a length less than 50 cm?

b.)Find the total distance traveled by the pendulum until it comes to rest.

Since the nth swing has length

100*0.99^(n-1), you want to find n when

100 * 0.99^(n-1) < 50
.99^(n-1) < .5
(n-1) log.99 < log.5
n > 69.9

So, on the 70th swing the arc is less than 50 cm.

Now you know the parameters for the geometric sequence. Just use the sum formula to add up 70 terms

To solve this problem, we can use the formula for the n-th term of a geometric sequence:

an = a * r^(n-1)

Where:
an = the n-th term of the sequence
a = the first term of the sequence
r = the common ratio of the sequence
n = the term number

a.) On which swing will the length first be less than 50 cm?

In this case, the first term (a) is 100 cm, and the second term is 99 cm. We can find the common ratio (r) by dividing the second term by the first term:

r = 99 cm / 100 cm = 0.99

Now, we need to find the term number (n) for which the length is less than 50 cm:

50 cm = 100 cm * 0.99^(n-1)

Divide both sides of the equation by 100 cm:

0.5 = 0.99^(n-1)

To solve for n, we need to take the logarithm of both sides:

log(0.5) = log(0.99^(n-1))

Using the logarithmic property log(x^y) = y * log(x), we can rewrite the equation as:

log(0.5) = (n-1) * log(0.99)

Now, divide both sides of the equation by log(0.99):

(n-1) = log(0.5) / log(0.99)

n = 1 + (log(0.5) / log(0.99))

Using a calculator, we find that:

n ≈ 68.97

Therefore, the length will first be less than 50 cm on the 69th swing.

b.) To find the total distance traveled by the pendulum until it comes to rest, we need to calculate the sum of the geometric sequence. The formula for the sum of a geometric sequence is:

Sn = a * (1 - r^n) / (1 - r)

Where:
Sn = the sum of the first n terms of the sequence

Using our values of a = 100 cm and r = 0.99, and assuming the pendulum swings indefinitely, we can plug these into the formula:

S = 100 cm * (1 - 0.99^n) / (1 - 0.99)

As n approaches infinity, the term 0.99^n approaches zero. Therefore, the sum (S) approaches:

S ≈ 100 cm * (1 - 0) / (1 - 0.99)

S ≈ 100 cm / 0.01

S ≈ 10000 cm

So, the total distance traveled by the pendulum until it comes to rest is approximately 10000 cm.

To find the swing on which the length first becomes less than 50 cm, we need to determine the common ratio of the geometric sequence.

Let's start by examining the lengths of the swings. We are given the first swing's length as 100 cm and the return swing's length as 99 cm.

In a geometric sequence, each term is obtained by multiplying the previous term by a constant factor called the common ratio (r).

So, we have the following information:

First swing: 100 cm
Second swing: 99 cm
Third swing: a_3 (unknown)
Fourth swing: a_4 (unknown)
...

Since the length of each swing decreases due to friction and air resistance, the common ratio (r) should be less than 1.

We can find the common ratio by taking the ratio of the second swing length to the first swing length:

r = 99 cm / 100 cm = 0.99

Now that we know the common ratio, we can use it to determine the length of each swing.

a.) To find the first swing with a length less than 50 cm, we can set up and solve the following inequality:

100 cm * r^n < 50 cm

r^n < 0.5

Taking the base-10 logarithm of both sides, we get:

log10(r^n) < log10(0.5)

n * log10(r) < log10(0.5)

n > log10(0.5) / log10(r)

Using a calculator to evaluate log10(0.5) ≈ -0.301 and log10(r) ≈ -0.004, we have:

n > -0.301 / -0.004

n > 75.25

Since the number of swings (n) must be an integer, the first swing with a length less than 50 cm will be the 76th swing.

b.) To find the total distance traveled by the pendulum until it comes to rest, we can utilize the formula for the sum of a geometric series:

S = a * (1 - r^n) / (1 - r)

where S is the sum of the series, a is the first term, r is the common ratio, and n is the number of terms.

In this case, since the pendulum comes to rest when the length is less than 0.01 cm (approximately), we can set up the following equation:

100 cm * r^n = 0.01 cm

r^n = 0.01 cm / 100 cm

r^n = 0.0001

Taking the base-10 logarithm of both sides, we have:

log10(r^n) = log10(0.0001)

n * log10(r) = log10(0.0001)

n = log10(0.0001) / log10(r)

Using a calculator, we find:

n ≈ 776.60

Now we can substitute the values into the formula for the sum of a geometric series:

S = 100 cm * (1 - 0.99^776.60) / (1 - 0.99)

Using a calculator, we obtain:

S ≈ 10,000 cm

Therefore, the total distance traveled by the pendulum until it comes to rest is approximately 10,000 cm.